History of calculus

Overview

Though the origins of integral calculus are generally regarded as going no farther back than to the ancient Greeks, there is evidence that the ancient Egyptians may have harbored such knowledge amongst themselves as well (see Moscow and Rhind Mathematical Papyri). Eudoxus is generally credited with the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into recognizable shapes. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See Archimedes on Spheres & Cylinders.) It was not until the time of Newton that these methods were made obsolete.

In India, the mathematician-astronomer Aryabhata in 499 used infinitesimals and expressed an astronomical problem in the form of a basic differential equation. Manjula in the 10th century elaborated on this differential equation in a commentary. This equation eventually led Bhaskara (1114-1185) to conceive of differential calculus and a number of ideas that are foundational to the development of modern calculus, including the earliest use of the "derivative" and differential coefficient, and the first statement of the idea now known as "Rolle's theorem". This theorem is important as a special case of the mean value theorem, which is one of the most important theorems in modern analysis. Using these concepts, he solved Aryabhata's differential equation to find the differential of the sine function, as well as the Earth's velocity in successive positions of its elliptical orbit around the Sun.

The 14th century Indian mathematician Madhava of Sangamagrama, along with other mathematicians of the Kerala School, studied mathematical analysis, infinite series, power series, Taylor series, trigonometric series, convergence, differentiation, integration, term by term integration, numerical integration by means of infinite series, iterative methods for solutions of non-linear equations, tests of convergence, the concept that the area under a curve is its integral, and the mean value theorem, which was later essential in proving the fundamental theorem of calculus and remains the most important result in differential calculus. Jyestadeva of the Kerala School wrote the first differential calculus text, the Yuktibhasa, which also includes discoveries of integral calculus, and explores methods and ideas of calculus that were later repeated, probably independently in Europe during the 17th, 18th and 19th centuries. These contributions likely laid the groundwork for contributions by Descartes and Fermat, which in turn led to the developments of Newton and Leibniz. Some historians have suggested that the contributions of the Kerala School to calculus were transmitted to Europe, but this is not known for certain. [http://www.canisius.edu/topos/rajeev.asp (See Possible transmission of Kerala mathematics to Europe.)

In 17th century Europe, Isaac Barrow, Pierre de Fermat, Blaise Pascal, John Wallis and others are said to have discussed the idea of a derivative. René Descartes introduced the foundation for the methods of analytic geometry in 1637, providing the foundation for calculus later introduced by Isaac Newton and Gottfried Leibniz, independently of each other. Fermat, among other things, is credited with an ingenious trick for evaluating the integral of any power function directly, thus providing a valuable clue to Newton and Leibniz in their development of the fundamental theorems of calculus. James Gregory was able to prove a restricted version of the second fundamental theorem of calculus.

Around the same time as Newton and Leibniz, there was also a great deal of work being done by Japanese mathematicians, particularly Kowa Seki. ^* He made a number of contributions, particularly in methods of determining areas of figures using integrals, extending the work of Archimedes. While these methods of finding areas were made largely obsolete by the development of the fundamental theorems by Newton and Leibniz, they still show a sophisticated knowledge of mathematics existed in 17th century Japan.

Newton and Leibniz are usually credited with the invention, independently of one another in the late 1600s, of differential and integral calculus as we know it today. Their most important contributions were the development of the fundamental theorem of calculus. Also, Leibniz did a great deal of work with developing consistent and useful notation and concepts. Newton was the first to organize the field into one consistent subject, and also provided some of the first and most important applications, especially of integral calculus. Of course, important contributions were also made by Barrow, Descartes, de Fermat, Huygens, Wallis and many others.

Development of Calculus

Some of the results of Newton and Leibniz were known to mathematicians in Kerala, India almost 300 years previously. In 1835, Charles Whish published an article in the Transactions of the Royal Asiatic Society of Great Britain and Ireland, in which he claimed that the work of the Kerala school "laid the foundation for a complete system of fluxions." It was not until the 1940s however, that historians of mathematics verified Whish's claims, proving that the Kerala school developed much of differential calculus well before Newton or Leibniz. Some historians propose these ideas may have been transmitted to Europe by the 17th century on the basis of methodological similarities, communication routes and a suitable chronology for transmission but there has been no evidence of direct transmission of manuscripts.

Historically, there was much debate over whether it was Newton or Leibniz who first "invented" calculus. This argument, the Newton v. Leibniz calculus controversy between the German Leibniz and the English Newton, was at the heart of a rift in the mathematical community of the two countries. Much of the credit for the resolution goes to the Analytical Society.

Much of the controversy centers on certain early manuscripts of Newton's that Leibniz may have had access to. Newton began his work on calculus at least as early as 1666, giving plenty of time for this to occur, as Leibniz did not begin his work until 1676. Leibniz was in England in 1673 and 1676, and probably did see some of Newton's manuscripts. It is not known how much this may have influenced Leibniz. Both Newton and Leibniz claimed that the other plagiarized their respective works. In fact, they may very well have influenced one another, but it is now widely accepted that the two developed their ideas mostly independently.

The work of both Newton and Leibniz is reflected in the notation used today. Newton was responsible for the $\dot{f}(x)$ and the $f'\left(x\right)$ notations, both very common in physics and mathematics. Leibniz was responsible for the $\frac{d}{dx}$ notation that is also very popular, particularly for problems in multivariate calculus.

Rigorous foundations

Calculus was widely used, as it was a very powerful mathematical tool, but it was not until the mid-1800s that it was put on a rigorous foundation from the modern viewpoint. For example, while the definition of the derivative itself has not changed since it was first introduced, it requires the notion of a limit. Newton, Leibniz, and their immediate successors interpreted limits intuitively instead of through precise definitions. This was standard practice at the time. Later, with the work of mathematicians like Augustin Louis Cauchy, Bernard Bolzano, and Karl Weierstrass, the foundations of calculus were clarified and made precise. The study of foundations eventually resulted in deep explorations of the concept of infinity by Georg Cantor and others.

Integrals

Niels Henrik Abel seems to have been the first to consider in a general way the question as to what differential expressions can be integrated in a finite form by the aid of ordinary functions, an investigation extended by Liouville. Cauchy early undertook the general theory of determining definite integrals, and the subject has been prominent during the 19th century. Frullani's theorem (1821), Bierens de Haan's work on the theory (1862) and his elaborate tables (1867), Dirichlet's lectures (1858) embodied in Meyer's treatise (1871), and numerous memoirs of Legendre, Poisson, Plana, Raabe, Sohncke, Schlömilch, Elliott, Leudesdorf, and Kronecker are among the noteworthy contributions.

Eulerian integrals were first studied by Euler and afterwards investigated by Legendre, by whom they were classed as Eulerian integrals of the first and second species, as follows:

\int_0^1 x^{n-1}(1 - x)^{n-1}dx

\int_0^\infty e^{-x} x^{n-1}dx

although these were not the exact forms of Euler's study. If n is an integer, it follows that

\int_0^\infty e^{-x}x^{n-1}dx = (n-1)!

, but the integral converges for all positive real

n

and defines an analytic continuation of the factorial function to all of the complex plane except for poles at zero and the negative integers. To it Legendre assigned the symbol

\Gamma

, and it is now called the gamma function. Besides being analytic over the positive reals,

\Gamma

also enjoys the uniquely defining property that

\log \Gamma

is convex, which aesthetically justifies this analytic continuation of the factorial function over any other analytic continuation. To the subject Dirichlet has contributed an important theorem Liouville, 1839), which has been elaborated by Liouville, Catalan, Leslie Ellis, and others. On the evaluation of

\Gamma (x)

and

\log \Gamma (x)

Raabe (1843-44), Bauer (1859), and Gudermann (1845) have written. Legendre's great table appeared in 1816.

Symbolic methods

Symbolic methods may be traced back to Taylor, and the analogy between successive differentiation and ordinary exponentials had been observed by numerous writers before the nineteenth century. Arbogast (1800) was the first, however, to separate the symbol of operation from that of quantity in a differential equation. François (1812) and Servois (1814) seem to have been the first to give correct rules on the subject. Hargreave (1848) applied these methods in his memoir on differential equations, and Boole freely employed them. Grassmann and Hermann Hankel made great use of the theory, the former in studying equations, the latter in his theory of complex numbers.

Calculus of variations

The calculus of variations may be said to begin with a problem of Johann Bernoulli's (1696). It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but Euler first elaborated the subject. His contributions began in 1733, and his Elementa Calculi Variationum gave to the science its name. Lagrange contributed extensively to the theory, and Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. To this discrimination Brunacci (1810), Gauss (1829), Poisson (1831), Ostrogradsky (1834), and Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), Hesse (1857), Clebsch (1858), and Carll (1885), but perhaps the most important work of the century is that of Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation.

Applications

The application of the infinitesimal calculus to problems in physics and astronomy was contemporary with the origin of the science. All through the eighteenth century these applications were multiplied, until at its close Laplace and Lagrange had brought the whole range of the study of forces into the realm of analysis. To Lagrange (1773) we owe the introduction of the theory of the potential into dynamics, although the name "potential function" and the fundamental memoir of the subject are due to Green (1827, printed in 1828). The name "potential" is due to Gauss (1840), and the distinction between potential and potential function to Clausius. With its development are connected the names of Dirichlet, Riemann, Neumann, Heine, Kronecker, Lipschitz, Christoffel, Kirchhoff, Beltrami, and many of the leading physicists of the century.

It is impossible in this place to enter into the great variety of other applications of analysis to physical problems. Among them are the investigations of Euler on vibrating chords; Sophie Germain on elastic membranes; Poisson, Lamé, Saint-Venant, and Clebsch on the elasticity of three-dimensional bodies; Fourier on heat diffusion; Fresnel on light; Maxwell, Helmholtz, and Hertz on electricity; Hansen, Hill, and Gyldén on astronomy; Maxwell on spherical harmonics; Lord Rayleigh on acoustics; and the contributions of Dirichlet, Weber, Kirchhoff, F. Neumann, Lord Kelvin, Clausius, Bjerknes, MacCullagh, and Fuhrmann to physics in general. The labors of Helmholtz should be especially mentioned, since he contributed to the theories of dynamics, electricity, etc., and brought his great analytical powers to bear on the fundamental axioms of mechanics as well as on those of pure mathematics.

Calculus History of mathematics

Histoire du calcul infinitésimal